Here's what I understand:
This is a 7-week CR to keep the gov open while they continue to hammer out appropriations bills for an actual FY2026 budget. The CR keeps spending at current levels - the R side didn't add any of their new priorities in there.
The D side is wanting to add new funding for over $1T in spending, including things that were included in the BBB as well as things that were recently taken away in the rescissions bill over the past month or two. Instead of continuing to negotiate those things in committees, they're trying to use this 7-week stopgap bill (which, again, doesn't add any additional money for either side's new priorities) to force new spending.
FYI: "Free healthcare for illegals" was something addressed in the BBB (adding new requirements to verify immigration status, prevent states from using federal funds to provide Medicaid benefits to undocumented immigrants, penalize states that use their own funds to cover non-citizens who are otherwise ineligible for Medicaid). These are the types of things the D side is wanting for their vote on the CR (along with the things aggiepanic95 stated above - those weren't jokes).
ETA: what Schumer is trying to push in public is extension of Obamacare subsidies that were passed in 2021 during COVID, which are set to expire at the end of the year. I believe those subsidies were passed with only D support, and they set them to expire at the end of this year. The R side's position is that COVID is over, the subsidies are no longer needed, and Obamacare is a total disaster that didn't garner a single R vote and if it's too expensive and requires subsidies to make it less unaffordable, the only option is to repeal/replace.
They're trying to cram a lot into this stopgap CR that belongs in the committees by holding the shutdown over everyone's heads. They are overplaying their hand here, as the Rs welcome the shutdown at this point.