What stops an attacker from crippling the Bitcoin network?
...could conceivably broadcast a ton of junk/illegitimate transactions to the network thereby consuming the necessary bandwidth and processing power for the network to deal with legitimate ones.

If the transactions are invalid, then they would not be relayed by the network's nodes, making the attack somewhat ineffective. At best, the attacker could consume computational resources of peers they are directly connected to, though if they are spamming invalid transactions then they would probably be banned by their peers soon.
If the transactions are valid, then the attack becomes expensive, which is a disincentive over the long term. It would be extremely expensive to fill every block with 'spam' transactions, and there will still be other users that will compete (in terms of fees) to have their transactions confirmed first.
If some entity attacked bitcoin via IP/DNS blocks, this would not entirely shut down the network, though it would likely cause disruptions to some nodes. I believe that nodes which connect through Tor, or through other methods (such as by satellite) would still function without issue.