First, a bit of background about how Congress gets its health care. Members of Congress and their staff used to receive their health insurance through the Federal Employees Health Benefits program, like many other government employees (this is all laid out in CRS report R43194 if you have some strange burning desire to know even more about lawmakers' health plans). Like many who receive health insurance through their jobs in the US, a significant portion of the premiums were paid for by the federal government (the "employer contribution"). One provision of the ACA (Obamacare) changed that by requiring Members and certain Congressional staffers to get their health insurance through the ACA's exchanges instead; this was intended to ensure that those in Congress got similar benefits as their constituents, at least in theory (Members also have the Office of the Attending Physician and can receive free outpatient care at military facilities).
When the ACA's exchanges started up, the Office of Personal Management (OPM) created rules to determine how health coverage for Members of Congress and staff would work. These individuals can buy special plans on the Washington DC Small Business exchange (DC SHOP), and if they do so, the federal government will make an employer contribution to cover much of the cost (but not more than 75%). They receive their plans through the exchange, but receive a special subsidy, just as their health care plans were subsidized prior to the ACA.
While this arrangement has been decried by some right-wing groups (Heritage Action in particular) as special treatment for Congress, the rule preserved the status quo, where the government paid the majority of premiums, and it allows Congress to hire and retain qualified staff, as other employers where staff are likely to work will usually offer health insurance benefits. Congress receives the same health insurance subsidy as other federal workers.
The rule that provides these subsides to Members of Congress and their staff is an OPM rule, and it's possible that the President could simply direct OPM to rescind it. While nobody knows exactly what his tweet means (a not uncommon problem), this is the most likely interpretation of "BAILOUTS for Members of Congress will end very soon". This would require Congress to immediately pay the full unsubsidized cost for their insurance, which would certainly unilaterally hurt them financially.
Whether this poses legal problems, perhaps because of the Twenty-seventh Amendment, is a novel question that would have to be addressed, along with the political implications of the President causing Congress to suddenly receive smaller paychecks.
Director of the Office of Management and Budget (OMB) Nick Mulvaney appeared to confirm this is what the President's tweet meant in an interview on CNN this morning, though he didn't answer how this would apply to Congressional staff as well (July 30, 2017):
Look, if Obamacare is hurting people, and it is, then why shouldn't it
hurt insurance companies, and more importantly for this discussion,
Members of Congress? There is a certain benefit that Members of
Congress get as part of an OPM decision from a couple of years ago and
I think the President's simply looking at this and going "is this
fair?"
[...]
Members of Congress are obliged by law to participate in the exchanges. I did when I was a Member of Congress. But there was also a decision from the Office of Personnel Management a couple years ago that allowed a special exemption to the rules on employer contributions to those plans. So it's not a concept of taking coverage away. The approach is actually obliging Members of Congress to follow the exact law that the folks they govern are following.
