Remember, in terms of physical fitness, 'bigger' and 'stronger' are not synonymous. You can get stronger without getting much bigger, and vice versa.
To answer your title question though, high intensity exercises altogether stop you from getting bigger and/or stronger (for the rest of this answer, I'll focus on 'bigger' rather than stronger). However, they do cause an increase in metabolism, and if you're not keeping up with the demand by taking in more calories than you burn on any given day, then your body will make up the difference by taking from your reserves. The thing to watch out for in the case of doing regular HIIT is that HIIT typically spikes your metabolism for a long time after the training, so just adding a handful of legumes or nuts to your regular diet may not cut it to gain weight, depending on your circumstance.
The good news, however, is you can eat as much as you like if you want to gain weight, just stay away from the sweets! From the thin people I've run across, they're always surprised at how much they have to eat to gain weight! I've never had that problem so I can't relate...
Before you start your training, I'd advise you to keep a diet log of everything you eat for a period of time (1-2 weeks should do), and get a good caloric estimate per day of what you take in. There are Base Metabolic Rate calculators out there that will give you a decent estimate of what you burn per day, without doing any work. The # they spit out isn't set in stone, but the actual rate should be pretty close.
From there, add a minimum 500-1k calories to your diet per day, every day, if you want to get bigger. If it's easier, you can start with an extra 250-500, and work your way up every day.
Now you have a baseline for your hiit also. Once you start jumping rope, you might find yourself more hungry even after eating your extra calories. This is your metabolism speeding up. At this point, you can choose to keep your calorie intake the same, or take in more to keep your mass the same (or even increase your mass further), it's up to you.
Finally, none of this will happen overnight. Do not expect to demolish a large pizza one day and be your dream size the next day, it doesn't work like that lol.