There are many reasons to use "Well" at the beginning of a sentence. One reason is to defend yourself.
When someone has attacked your actions or something that you said, you can say "Well" before you explain yourself:
A: You never do the dishes!
B: Well, that's because I don't use as many dishes as you.
A: You broke it!
B: Well you're the one who told me to hold it like that!
Defending yourself with "Well" can make you sound a little childish when you use it in an argument. But you can also use "Well" to defend yourself a little more calmly:
A: Don't you have to go to work soon?
B: Well, yeah, but it's OK if I get to the office a little late.