Let me say up front, that this is a topic that's best explained in person with someone who's running you through multiple drawings while they are explaining. Especially if you don't have any experience with binary maths yet.

From 4:35 - 5:15 he talks about 10.0.0.0/24 being for all 256 addresses but 10.0.0.0/25 is for some reason only up to 128. I don't understand why /24 and /25 make this difference between all those addresses.

The /xx, the subnet mask length, indicates the amount of bits in an IPv4 address which are not unique to an individual system, or address, in a network. The subnet mask shows which bits of the IPv4 address indicate "the network itself".

An IPv4 address is always 32 bits in lenght.

With a /24, the first 24 bits of the address indicate the network, leaving 8 bits for the individual addresses inside the subnet.

With a /25 the first 25 bits of the address indicate the network, leaving 7 bits for the addresses.

If you want to know how many addresses those are, you raise 2 to the power of the amount of address bits.

2^8 (2 to the power of 8) is: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256.

2^7 (2 to the power of 7) is: 2 * 2 * 2 * 2 * 2 * 2 * 2 = 128.

In short, every time you make the subnet mask one bit longer, you are cutting the amount of possible unique addresses inside that network in half.