In a nutshell: In mindfulness practice, you develop skills to focus on the present, real-world moment, which can help manage feelings of overwhelm, anxiety, and depression. Using these skills regularly can help preventing these feelings from building up.

What does “mindfulness practice” even mean?
I know. The term “mindfulness” is everywhere these days. It seems abstract, trendy. Like it may require you to have a special bell, or a mat.
There are various ways to practice being present in any given moment. In some, you tune into sensory input—what you see, hear, feel, and so on. Others emphasize meditating on your breathing or other images (guided by someone else or on your own. And some physical exercises (yes, yes, some of which include mats). But you can also practice mindfulness doing the dishes, taking a shower, and in conversations.
But it’s really very practical: it’s just about paying attention to what is here-and-now, and accepting whatever you find for what it is, even if you wish it were different.
Real-life benefits
Practicing mindfulness as a habit has remarkably concrete benefits over time. It can help manage overwhelming emotions, depressive feelings, chronic pain, anxiety, and sleep problems. Beyond that, it can make you aware of your experience—of the internal and external—in a way that is its own reward.
Learning mindfulness skills is a huge benefit in therapy, because the ability to both accept and manage feelings allows us to develop insight and make decisions based on our priorities.