Intro: don't expect it to work in your favor
First thing to keep in mind is that purpose of hot questions feature is established pretty firmly and that it's not really about helping in your community growth. The feature is there only to show network wide audience entertaining / interesting questions. If you're interested, refer MSE for more details on that: What is the Goal of “Hot Network Questions”?
Because of this, you should not generally expect this feature to work in favor of your community. You can refer examples at Programmers and Workplace to find more details about that. "Tons and tons of people visited the site... but very few decided to stick around..." "flash in the pan..." - that's what you can expect (unless you put dedicated effort into adapting it to your needs, but more on that below).
To avoid misunderstanding, above is not to discourage you from trying to use it for the site's advantage, nor even from requesting functional changes to it for the benefit of your community. It's only there to help you avoid incorrect assumptions and expectations about why the feature is there in the first place.
Now, assuming that you intend to leverage hot questions for the benefit of your community...
What you absolutely should do
- Edit everything that needs improvements in spelling and formatting.
Your site regulars might know that posts of user JoeTheExpert are well worth reading despite their habit to lowercase "I" and dump wall of text without paragraph breaks. But you are willing to attract outside visitors who have no idea about this and if a valuable content is hard to read for them, you risk losing them.
Keep in mind that typical visitors expect entertaining read when clicking hot questions, they simply are not prepared to put much effort into decoding obscure posts... and as newcomers, they have no way to know that JoeTheExpert is worth reading despite difficulties. If you want them read and decide to stick, make your best effort to help them see that your content is worth sticking.
- Be merciless about off-topic questions
Your site is about tulips, but sometimes people make mistake and ask about roses instead. You understand that and let these questions hang around for a while before they get migrated to roses.stackexchange. Some of your users may even shot an advice or two while it's there waiting for migration. If the question is well presented, some may even cast a sympathy upvote - why not, if the asker apparently put solid effort into crafting their question.
This is understandable and may work just fine in regular questions. Site regulars are well aware that roses are off-topic and that question will eventually get away anyway, no big deal.
But when that roses question gets into hot list, things change. You need to understand that while it's there, advertised to hundreds if not thousands outsiders, it sort of makes a loud statement that your site is about roses.
- Hot list readers aren't your site regulars, they have no way to know that it's off-topic. Seeing this question will make them think that it's okay to ask about roses here. And don't expect that "tulips" in site URL or "this site is not about roses" comments from site regulars could help - inexperienced visitors won't pay attention to these subtle hints.
If you don't want for hundreds newcomers to fall into thinking that your site is about roses, try to do your best to drop this question off the hot list. For that, you first need to understand what keeps it there. That's really simple: question must be open to begin with (closed questions drop off the list) and the higher is the score of question and answers and the more answers are there, the better chances for it to get and stick in the hot list.
So, if you want to drop the off-topic question from the hot list, 1) try to get it closed sooner, 2) vote down and 3) abstain of answering. For the latter, additional feature to consider is protection from new users' answers but more on that later.
Beware of answer quality problems
High exposure of the question to the inexperienced visitors tends to bring answer quality issues:
Amount of views and contribution of newcomers... much increases, along with the risk of things getting out of control.
You know, everyone would want to answer a question, but for community regulars, this natural desire is balanced by established norms. Regulars understand what kind answers are good and likely to bring upvotes and they avoid posting garbage that usually gets downvoted: this sort of sets the quality norm.
But wait, the question is now on top, it attracts a lot of visitors whose desire to answer is not balanced by local community norms...
Stack Exchange team established a dedicated automatic protection feature intended to help with issues like this. If voting on your site is relatively anemic and if site doesn't get many visitors "armed" with association bonus reputation (like it is for example at Stack Overflow), expect basic protection offered by this feature suffice to save you from most troublesome effects of popularity. If voting at your site differs from that one you may find it a bit too fragile - I am not aware of general solution for that, you'd probably have to discuss it at site meta if it grows into substantial problem.
- Anyway, even in the presence of said auto-protection you may have to invest substantial extra effort into maintaining appropriate answer quality in popular questions. Auto-protection saves you from getting more of troublesome content but it is expected to trigger when some of it already leaked in - so you would have to take care of that yourself.
Consider actively editing, checking for repetitive and low quality answers, up to the point of involving moderators in cleaning up and deletion of troublesome content. A solid guidance on the latter was given a while ago by Jeff Atwood:
...answers that are strong candidates for deletion:
belong to low rep or anon users with no real commitment to the community
are provably duplicate, that is, were added well after (30+ mins later) other answers that contained the same exact information
are short in length
do not explain much of anything
Now, to the main point of your question...
How does this attract participation
...and not simply this sort of "drive-by viewing" of popular questions?
For that, you need to think one step ahead so to speak. Hot questions attract multiple random visitors, hundreds or even thousands of them...
...and if you want to leverage this "firehose of publicity", you better set your goals firm and straight.
Among all the visitors, you need to address those who may be genuinely interested in site topics (could be just one of 100 but it's them who you need to target) and convince them that your site is a place worth sticking around to serve their interest in your topics.
What is available to you is just that hot question and its answers, nothing else. You shouldn't realistically expect them to carefully browse and study other stuff, because at the moment of their visit your site is to them only a host of "that curious link at sidebar", nothing else.
- You need to convince your potentially valuable user that your site is worth sticking around. The only way for this is to present that question and its answers as an example of a topical, valuable, clearly laid out content. A Group Is Its Own Worst Enemy article explains what should be your goal here (made bold in the quote below):
...my favorite pattern is from MetaFilter, which is: When we start seeing effects of scale, we shut off the new user page. "Someone mentions us in the press and how great we are? Bye!" That's a way of raising the bar, that's creating a threshold of participation. And anyone who bookmarks that page and says "You know, I really want to be in there; maybe I'll go back later," that's the kind of user MeFi wants to have.
Technical details of MetaFilter site differ from these of Stack Exchange, but general idea still applies.
Of all the crowd brought in to hot question, you need to catch a few interested in your topics, make them bookmark your site and say, "You know, I really want to be in there; maybe I'll go back later".