Maybe a not-so-technical view helps understanding the problem...
Just think of an envelope that contains the message and the hashsum. You can punch the envelope and flip certain bits of the message. You can also punch where the hashsum is and change bits of it. However, as you cannot look inside the envelope you have (almost) no chance to flip the hashsum in a way that yields the correct value for the modified message.
When the receiver opens the envelope they see a message and a hashsum that do not match - a clear indication that something was modified, the message, the checksum or both.
Ok, more technical... :-)
Consider the following message M, which I transfer plain-text.
Please transfer 100$ to my account.
its (SHA1) hashsum is 5f7e22d270ca9da68543e3f97f30f1859e20a88a.
The resulting MAC is
This MAC is the hashsum that I encrypted with a secret password. You can freely modify the message and the MAC, but you cannot figure out how to modify the MAC so that, when I decrypt it, it delivers the correct hashsum, that matches your modified message. Consequently when I decrypt, I will notice that the hashsum does not match the message and something was modified.
That is what a MAC is all about, detecting modifications.