The following assumes that one understands what happens with a soft and hard fork. See all the answers here for a good explanation.
Say a new change is implemented in the bitcoin protocol and it requires a hard fork. Full nodes are required to upgrade to the new version (otherwise there are dangers...).
What are the dangers of a hard fork?
The real danger is that the blockchain will split in two parallel blockchains. Miners, merchants, users (via wallets and other software), etc. that did not upgrade will support/use blockchain A and miners, merchants, users, etc. that did upgrade will support/use blockchain B.
How would people lose money due to a hard fork?
Effectively there are now two bitcoin blockchains. The bitcoins that you had before the fork will exist in both blockchains A and B (they doubled!). A simplified example would be that: You can use your bitcoins from an old wallet and buy something from a merchant that uses old nodes. Then you can use a new wallet and buy something from a merchant that uses the new nodes with the same bitcoins. This is not even doublespend since the chains are different.
However, what this also means is that the bitcoin users and community are also split. Each blockchain instead of having X nodes (the nodes before the split) that keep it secure will have X/2 nodes. Several merchants accept bitcoin but some accept version A and some version B... and so on. This is not user friendly and it will cause much confusion. Something like that will shatter trust in the bitcoin network causing its price to diminish. Fear of future hard forks (splits) will diminish trust (and price) even more. So people will lose money indirectly since the value of bitcoin will collapse (best case scenario the price will split... but it is probably going to be much much worse).
Why is it necessary to give more than a month time to prepare for a hard fork?
Because we want to make sure that everybody is aware of the change; miners, merchants and users. We want to make sure that they do not have objections to the change and that they agree on the upgrade. In addition wallet creators and merchants might also need time to upgrade their software accordingly (depending on the upgrade changes). The delay is for both (first) social and (second) technical reasons.
Also note that if the split is, say 95% upgraded - 5% did not (or vice versa), it will not be a big problem since the bigger chain will have the overwhelming majority and the remaining will be forced to upgrade or be ignored.