Yes, it really does matter how often you change out your running shoes with a new pair! That sounds like a really nifty and helpful app, and you should probably take into consideration what it advises you. Because it can accurately track how far you have ran and keep up with how many miles you have logged over time, it can also accurately keep up with your total mileage ran in one pair of shoes.
Shoes are like cars. Over time, after many miles and a lot of wear and tear, they get worn down and do not work as well. We put our running shoes through a lot! We train in some rough conditions, expose them to some extreme weather conditions, and they get drug through rain puddles, snow storms, dirt, mud, grass, gravel, and all other kinds of nasty conditions. Over time and with repeated use, our shoes start to lose their comfort, stability, cushioning, and support. Their insoles get worn down and as a result, they do not offer our feat as much stability and protection. This can mean aching heels, arches, and toes, as well as prolonged and worsened injuries to ankles and knees.
A general rule of thumb that most runners agree on is to change out your shoes every 300 to 500 miles. Lean more toward changing them after every 300 if you like to hit the trails, do a lot of running outside, and also wear them doing other activities like errand running and weight lifting. You can get about 500 miles or so out of a pair of shoes that you save just for running, and if you do most of your running indoors on a treadmill or on cleaner, paved surfaces.
What kind and brand of shoe you have also plays a role in determining how long they will last. As with most things, more expensive running shoes are made to last a bit longer. If you want to get more serious about your running, and if you want your shoes to last longer, nicer running shoes are worth the investment. Check out shoe brands like Mizunos, Brooks, and Asics for some of the better running shoe options, and then be sure to go in-person to a specialty running or athletic store to try them on so that you know if they will work for you.