Chains of Mephistopheles {1B} |Enchantment| If a player would draw a card except the first one he or she draws in his or her draw step each turn, that player discards a card instead. If the player discards a card this way, he or she draws a card. If the player doesn't discard a card this way, he or she puts the top card of his or her library into his or her graveyard.
The card is a bit of a mouthful, and has always been. The replacement effect it creates is a bit more complicated than most. Instead of replacing one simple event with another simple event, this replaces a simple event (drawing a card) with an entire process.
If you would draw a card, you instead do the following. First you have to discard a card. This is not optional, if you have a card in hand, you must discard it. Next one of two things happens, based on whether you discarded a card or not. If you discarded a card, you then draw a card. If you didn't discard a card, you put the top card of your library in your graveyard.
So to simplify it down as much as possible, when instructed to draw a card, if you have a card in your hand, you discard then draw. If you do not have a card in your hand, you put the top card of your library into your graveyard.
Now we can look at Anvil of Bogardan. The first thing that happens during your draw step is you draw a card. Then Anvil's trigger goes on the stack. (Assuming nothing else happens before it resolves...) When the trigger resolves you carry out it's insturctions in order as printed. The first instruction is to draw a card, this however, gets replaced by Chains' replacement effect, and since you have a card in hand (the one you drew for your turn, at least), you discard a card then draw a card. Then you carry out the next instruction in Anvil's trigger, discard a card.
Simplified version of draw step: Draw a card, Anvil triggers, on resolution of trigger discard, then draw, then discard.