One reason why funding follows the markets is that venture capitalists themselves rely on funding from limited partners, the entities that invest in VC firms. LPs need to keep their portfolios balanced between public market securities, like hedge funds, and private markets, like venture capital.
When the stock market crashes, their venture capital asset class investments all of a sudden represent a larger percentage of their portfolio (as their stock market positions shrink in value), so they may be less willing to invest in a VC firm.
A VC firm that was going to raise its next fund might decide to slow the pace of investment to make the fund last longer.
VCs also think opportunistically, and when the total activity in venture drops, startups will demand less money and a lower valuation. So VCs may wait to invest in lower valuations.
Finally, most LPs require VC firms to invest over a span of a few years to get time diversity in the fund. The hot years are thus balanced by the cooler years.