Why is there water pooling under my furnace?

That damp hose going to the floor drain is almost certainly your answer. High efficiency furnaces pull a lot of water out of the exhaust gas, and all that moisture drains thru a small hose and usally a little trap before it hits the floor drain. When everything's working it just trickles along quietly, but if that line gets plugged or kinked, the water has nowhere to go and it backs up out of the collection box or weeps out around the connections. Right at the front bottom of the unit is exactly where it shows up on mine. The cold snap matters too. If that drain line runs anywhere near an exterior wall or through a cold part of the utility room, it can partially freeze and slow the flow down. Then it warms up, everything thaws, and you get a sudden release of water that pooled up in the meantime. I've had this happen twice, both times right after a hard freeze.

Easy way to check it yourself. Kill the power to the furnace at the switch, then follow that hose from the unit down to the drain. Pull it loose at the drain end and blow through it toward the furnace. If you get resistance or it won't pass air, there's your clog. Also check the floor drain itself, because they collect lint and dust and can be nearly sealed over. Pour a bucket of water in and see if it drains at a normal speed.

Sometimes the clog is actually in the little trap on the furnace side rather than the hose. That part usually lifts off or has a removable cap, and gunk in there is common. Flush it with warm water and reassemble. If you clear the line, dry the area up, and run the heat for a few hours without a new puddle, you've fixed it. If water shows up inside the cabinet around the collection box even with a clear drain line, then yeah, that's tech territory because it could be a cracked collector or a bad gasket, and that's not a cheap guess to make on your own.

That white PVC setup is why us old guys laugh a bit. We spent decades with cast iron furnaces that sent all that moisture right up the chimney, never saw a drop in the utility room. Now the units squeeze every last bit of heat out and you get a puddle for your trouble. Had one of those little drain hoses develop a pinhole years back where it sat against the cabinet edge. Looked bone dry up top, dripped out the bottom for weeks. Run your fingers along the whole length of the hose, not just the ends. Sometimes the connection at the furnace just needs the clamp snugged and a fresh cut on the hose end.

Seconding the drain line theory but with a bit of context that made me stop panicking when I first dealt with this, these units can produce a genuinely startling amount of water, mine will put out better than a gallon a day when it's really cold outside, so a dinner-plate puddle twice in one afternoon doesn't automatically mean something dramatic is failing. One thing I'd trace while you're at it is wheather the hose sags below the fitting anywhere, becasue a low spot holds a pocket of water that can block the airflow through the line even when there's no debris in it at all, and re-routing mine with one continuous downhill run ended a recurring leak for good.

Yeah, agreeing with the guy who mentioned the freeze and thaw cycle, that lines up with what happened here. Mine did the same thing one January, ran fine all day then I found water at the front and figured the worst. Turned out the drain hose had sagged where it looped behind the blower housing and made a low spot that held water, which then froze solid overnight. Re-routed it with a gentler slope and never had the problem again. So check the whole run for dips, not just the ends where it connects.