Then, note that each term here is scalar, and so $y^TXb$, i.e. ($(1\times n)(n \times k)(k \times 1)$), where $n$ is number of samples (data points), $k$ is number of regressors, including bias term.
For scalar terms, we can either take the transpose or not, i.e. $\alpha=\alpha^T$. So, $y^TXb=(y^TXb)^T=b^TX^Ty$, which is the third term above. Then, MSE becomes $y^Ty-2b^TX^Ty+b^TX^TXb$.
Note: If you give additional information about where you confronted this equation (e.g. linear regression lecture etc.), it'd better suit this forum; otherwise, this question may also suit to math forum (maybe better), although the procedure is quite common in ML for MSE calculation.