Suppose we have $n$ bits of random-looking data, and we want to encode it in such a way that instead of 1/2 the bits being 1's, we have (say) 3/4 the bits being 1's. The entropy of each bit in the new encoding is $-0.75\log_2(0.75)-0.25\log_2(0.25)\approx 0.81$, so we should be able to do it in about $1.23n$ bits.
(By 3/4 the bits being 1's, I mean that each bit should be indistinguishable from an event that happens with probability 3/4; as opposed to a long block of unbiased random bits followed by an equal length block of 1's.)
We could try, for each bit, put the next bit of the original data with probability 1/2, and put a 1 with probability 1/2. Then somehow include extra information about which bits encode the original data, and which are filler. But this reqiures at least $2n$ bits, well above the theoretical minimum. So how can the minimum actually be achieved?