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I'd guess it has something to do with reducing interest expense in addition to reducing debt.
Increases enterprise value by reducing net debt, hence higher return as there will be a higher EV exit multiple (thus higher ROI and IRR) at exit date. Sponsor can also use FCF to organically grow business to increase EV.
The decision to either accumulate cash or pay down debt over the hold period using said cash does not increase your exit multiple.
If interest expense was zero, your IRR would be indifferent between sweeping cash to pay down debt or accumulating cash.
The higher IRR you receive by paying down debt over the hold period is purely a function of reducing interest expense which, given its circular nature, generates more cash to pay down debt and hence you have a slightly lower net debt position at exit (increasing equity value and returns).
I think you missed the point of OP's question. More excess cash (hoarding cash) also decreases net debt. From the way the original question was worded, I think it's already understood that paying down debt is one way to increase the value to equity.
My stab at answering the question is that I don't think paying down debt increases value to equity more than hoarding cash does (besides the interest expense point, mentioned by a poster above). My understanding is that you pay down debt so you can stay within financial covenants, pay mandatory payments (If I understand correctly, sometimes cash sweeps are written into the docs as well, so you may be on the hook for more than just "mandatory" payments), stay on lenders' good side, avoid downgrades that make refi / reprice more difficult down the line, etc.
If you keep the cash on your BS, you earn little to no interest (say, 0.5%). But you still pay interest on your debt (in an LBO, think 4–8% depending on leverage).
If you pay down the debt, you earn less interest on available cash (missing out on a small cash-in) but you pay less interest on outstanding debt (avoiding a proportionately larger cash-out). Hope this clears it.
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