Lordy folks, it's not that hard to understand.

Earlier on in the movie, the tripods had an energy shield surrounding them which prevented any human weapons from being effective. Nothing could get inside the energy shield. Remember outside Tim Robbins' house when Tom Cruise throws the grenade at the tripod and it explodes before it hits? That's because it hit the shield.

In Boston, Tom Cruise sees the birds flying around and landing on the tripod. If the shield was there then the birds wouldn't be able to do that. They'd fly up to the shield and then get fried or something. Or at the very least bounce off. That's why when he's yellin to the soldier about "no birds!" he finally just says "no shield!" With the shield down then human weapons could hit the actual tripod and blow it up.

As for why the aliens died, it had nothing to do with the birds. Remember how in both the beginning and end of the movie you see a close up view of single celled organisms in a drop of water? The narrator explains at the end that the aliens died because they got sick from our diseases. Those bacteria and our immunity to them are what "give us the right to keep our planet." It's a play on how when Europeans invaded America, the natives died from diseases brought over from Europe. This time the invaders are the ones who die from native disease instead of the other way around.

Hope you guys get it now.