The rate used in a DCF model is the required rate of return for that investment. It implies that you have other options for where to put your money, and that there is an opportunity cost to committing funds.
Governmental interest rates represent the 'risk free' rate of earnings achievable as an alternative to the cashflow you are modelling. The rate used in a DCF should consider the risk-free rate + an appropriate modifier to account for the additional risk of your particular investment type. If a 7% rate is 'fair compensation' when T-Bills offer 2%, would you really still call 7% 'fair compensation' when T-Bills offer 2.5%?
If the underlying investment hasn't changed [and therefore, has the same risk profile and, simplistically, the same modifier to its required rate of return], then rising interest rates should have similar impacts to most DCF models.
In an extreme scenario, what happens if t-bills rise to 10? Would you really still be okay with 7% equity return [implying the cashflows you are viewing have a lower risk rating than the US government]?
So, required rate of return rises in reflection of the relative increase in the value of risk-free government bonds, the NPV of those cash flows decreases, and investors are no longer willing to pay the same amount to purchase them [the price drops].