Why does the background radiation in the universe have a temperature of 3ºc above absolute zero? i know it's a result of the big bang, but why is it specifically 3ºc? wouldn't the universe cool down if it was expanding?

The background radiation is associated with a temperature of about 2.73 Kelvins (not Celsius). There is no particular reason for it to be at this specific value. This temperature is a snapshot of an ongoing process taken at our point in time. This temperature continues to change from where it used to be - many orders of magnitude higher in the early stages of our universe model - to today's measured value. The expanding universe view implies this temperature continues to decline as the universe expands, as you pointed out. It is just happening on an extremely slow time scale that we consider this value to be a constant.

ι вєℓιєνє ιт ιѕ тнє fιяѕт ¢нσι¢є(α.)

нє ѕнσυℓ∂ ѕмιℓє ρσℓιтℓу αи∂ ѕρєαк тσ тнє ρєяѕσи ωнσ ιѕ ιитєяяυριиg αfтєя тнє ∂ιѕ¢υѕѕισи

нσρє ι ¢συℓ∂ нєℓρ.