"for four straight days" means something has been going on constantly for four days, for example a machine has been running for four straight days. "four days in a row" means that you have been doing something every day for four days, for example you had breakfast in the hotel four days in a row. It is grammatically correct to omit "for" and "the" in these sentence examples :)

You can find the connection between the two similar ways to state time passing by understanding the word straight. Straight means something linear, a line, a row. So you can see the usage is slightly different but they both come from the same conceptual root, one from Latin's and one from Saxon's root.