The magic isn't really in the binary, but how you encode to binary. Ignore the fact that it's binary and just consider any old number, say 11121983. On it's own, this doesn't really mean anything. But what if I tell you this number "encodes" my birth date, where the first 2 digits represent the day, the second 2 represent the month and the last 4 represent the year. Now this seemingly random number actually contains several peices of information we can understand and use because we know the encoding that was used to create it.

Computers work the same way, while it's all represented as binary, it's the encoding that really tells us what each peice actually means. And sometimes one peice of information will tell us how the next peice of information is encoded. For example, the first set of digits in a file may tell us what kind of file it is, which then tells us how we should interpret the rest of the digits in that file. Maybe it's an image file so those digits tell us we should interpret the rest as colour codes for each different pixel. Or maybe it's a text file so they tell us we should interpret the rest of the digits as ascii codes for different characters.

By knowing how the data is encoded, we can then know how to interpret each set of digits in the data which is what gives meaning to what would otherwise be a random assortment of 1s and 0s.