To dip into means to have the need to use something (usually something of great value). It's like being forced to remove the first few slices off a loaf of bread (a very important loaf of bread). This usually means having to utilize an asset that you didn't plan on needing to spend. For example, if you have a savings account that's just used for fun things like vacations, you wouldn't like to use this account for other (not so fun) things. Emergency funds, such as in your example, are something you never want to waste. The best example I can give: "When my son got arrested and needed bailed out of jail, I had to dip into the funds that were supposed to be used for my retirement to free him."

To answer your second question: another word that can be used... You can just say "spend". For instance, "to go on vacation, I'd have to spend some of my emergency money, and I just don't want to do that." "Dip into" is slang but a more formal word would be choices like "spend" or "borrow". For instance: "To go on vacation, I'd have to borrow from my savings, and I just don't want to do it."

"Dip into" relates to using or indulging, often in a casual way. In this case, the speaker is considering taking out money from savings, never a good idea.