Why are birds closer to reptiles than to mammals?

It has to do with the evolutionary tree. Shortly after the first reptiles appeared they split into synapsids and sauropsids. Mammals eventually branched off from the synapsid reptiles. After synapsid reptiles went extinct, mammals were left as the only synapsids.

Meanwhile the sauropsids diversified into a number of different groups including dinosaurs and the ancestors of all modern reptiles. Later on birds emerged as a branch of the dinosaurs.

In summary: mammals are synapsids while birds and modern reptiles are sauropsids.