How does a butcher age meat for a week while It doesn't keep in my fridge for more than three days?

Meats aren’t like wine, you don’t want them aged. Generally you want them as fresh as possible. Certain, very select, cuts of beef are stored for a certain period of time (generally a month to a month in a half) to achieve a certain quality of texture. But, again, these are very select (in terms of fat distribution) cuts and require storage in conditions you’re not really going to be able to replicate in your apartment. These cuts are expensive and you don’t typically find them outside of high-priced steakhouses and, maybe, high-end, gourmet meat shops that cater to wealthy clientele.

The stuff you, I and pretty much everyone else gets at the grocery store is as aged as it needs to be by the time it reaches your shopping cart.

The grocery store I go to has a special glass dry-aging container for very large pieces of meat.

You can age your own meats, but you need to buy very large cuts (10 -15 lbs. minimum) and then store it in a dehumidified cooler. And, as @Darth_Algar said, it is for expensive cuts, Prime grade. It is best for meats that will cook quickly, like a steak, not for roasting or stewing.

To add to what jellies wrote above, I don’t think you’ll want to do it. In my memory mold grows on the beef as it ages, and then that is cut away before it is sent to the restaurants to prepare. Again, these are specific premium cuts of meat that are dry aged for a long time. The aging process shrinks the meat, so that partly accounts for the much higher price.

Your refrigerator may be set too warm. Check it. It should be about 40 degrees, if my memory is correct.
Everyone else here who says that special dry dehumidified storage spaces are required to age meat are correct. Again, if memory serves me correctly.

Here’s all you ever wanted to know about aging beef, and why it’s different than doing it in your own refeigerator.

https://en.wikipedia.org/wiki/Beef_aging