So traditional Venture firms are not part of this (1). VC buys equity in their start-ups, not debt, so interest rates aren't a direct concern of theirs. And VC firms generally do not borrow, they raise investment funds of investors money and because it is high risk, generally speaking no one lends money to these funds.

Private Equity firms on the other hand do generally take on a lot of debt- they have their own money but they borrow even more (2) and use that amount of money to buy out the current owners of a company. Basically they borrow to turn a steady stream of revenue into the future (an existing software company that is profitable) into a big pile of cash today for the current owners. And then they need to pay off that borrowing over time, from the steady stream of revenue that the software company generates. And if they are wrong about the size of the steady stream of revenue, they basically declare bankruptcy and the people who loaned the money (banks but also pension firms etc.) now run the company.

1: The way that VC changes is that the rising interest rates mean that the returns to safe investments are higher, so in order to attract new capital VC firms have to promise even higher returns as well.

2: They borrow money to get higher returns- if you have 5 dollars in a stock and it goes up 20% you have 6 dollars. If you have 5 dollars and borrow 5 dollars and invest it in a stock that goes up 20% you have 7 dollars after you pay back the loan. The catch is, if the stock instead goes down by 20%, after you pay back the loan you only have 3 dollars. (And if the stock goes down 50% you are wiped out.) But PE firms are confident that their investments will always be winners, so they take out loans like that.