This question raises a lot of interesting problems about the continuity of self and consciousness, but I think it can be cut down with a very pragmatic, down to earth approach.
Quite simply, the idea that the person we will be tomorrow is a different individual than the present us is contradicted by the fact that, so far in our life, we have the continuous experience of having to live with the consequences of our past actions.
Think about any decision you made that you regretted. I bet you experienced this regret and hardship first hand, subjectively, as happening to you, not to a third person. This feeling of regret, coupled with the memory of making the decision(s) that led to it, constitutes the experience that allows us to figure that, should we take a similar decision in the future, we will have to experience first hand the consequences, not some distant and abstract "future me".
Our experience tells us that if we party too hard on Saturday, we are hung over on Sunday. Whatever word game we can make up that "tomorrow's us is not us", the fact is that we, personally, in the first person, have to suffer through the headache and nausea. This is what we experience to be the case. It simply never happens that we party, and someone else has to go through our hangover.
As we grow older and gain experience, we are able to infer this pattern over longer time periods and more abstract problems, like saving money or keeping healthy. We know that if we don't save now, we, and no one else, will have to endure the state of being broke because of it. It is what our whole experience of life indicates.
Of course, this has to be mitigated with our desire to enjoy life here and now. It becomes a straightforward trade off: how much sacrifice now is worth avoiding trouble later. This is an emotional balance between our desire to enjoy now and our desire to avoid foreseeable trouble, directly conditioned by our ability to accurately forecast said future. (Concerning the balance of desires in the individual guiding our choices, see Spinoza's Ethics). Modern psychological studies tend to support the idea that our decisions are motivated by emotions first, and only then supported by a rational argument we make up for them.