How does one end up with a non-integer exponent in time complexity? (e.g., $O(n^{2.81})$)
The running time complexity of Strassen's algorithm is given by the recurrence
$$
T(n) = 7T(n/2) + O(n^2).
$$
(With a suitable base case.) The solution of this recurrence is $T(n) = O(n^{\log_2 7})$.

Strassen's algorithm multiplies two $n \times n$ matrices $A,B$ by decomposing them into four $(n/2)\times(n/2)$ matrices each, computing seven linear combinations of the smaller matrices each, say $(A_i,B_i)_{i=1,\dots,7}$, recursively computing $C_i = A_i B_i$, and computing the four $(n/2)\times(n/2)$ matrices of the result by taking linear combinations of the matrices $C_i$. That's how this running time came to be. If you want to learn more, there is a lot of information out there on Strassen's algorithm. There are, by the way, asymptotically faster algorithms for matrix multiplication, the current champion being Le Gall.