Have you ever played with blocks before? Imagine that you have a bunch of different colored blocks and you want to figure out how many of each color you have. Bayesian regression is like taking all of the blocks and sorting them into different color piles.
But, sometimes you might not have enough blocks to make a really good guess about how many of each color there are. Bayesian regression helps you make a guess by using information from other things you might know about the blocks or the colors.
Imagine you also have a box where you can draw out blocks without looking at them. You don't know what color you're going to get, but you can still use this box to help you make a better guess about how many of each color you have. If you draw out more blue blocks, then you can make a better guess that you have more blue blocks overall.
Bayesian regression helps you do this for lots of different things, not just blocks. It can help you make better guesses about things like how fast a car is going or how likely it is to rain tomorrow. By using information from lots of different sources, you can make a pretty good guess even if you don't have all of the information you need.