I suspect that you understand all of the words and phrases but want some information about the background?
In a nut shell (this is an informal expression meaning that one is about to express a summary of something -- in this case something complicated), the situation is as follows:
In the UK we have something called the National Health Service (NHS). It is publicly funded. This means that everybody that earns a certain amount of money from their job has to pay taxes to the government to fund the NHS.
Over time, as is the case in other industrialized countries, our people are living longer and longer. This is mainly because of advancements in medical science - for example some forms of cancer are becoming so treatable that they can now be considered chronic illnesses as opposed to things that are likely to kill you.
But as people are living for longer periods of time, then more of them are living to such an age that they develop chronic conditions - such as dementia - which make them (the people) very expensive to care for. And if the number of people that are working and being paid wages remains roughly the same, then there won't be enough money to pay for the care of all of these extra old people.
Therefore, the Government has decided that everybody should be made to work up to an older age before they can retire on what we call the State Pension (money paid by the state to old people to allow them to live).
The key point being made in this passage is that if those people that are forced to work longer get ill or become injured as a result and have to be treated on the NHS then the logic behind this policy will fall down (the phrasal verb "to fall down," in this context means to fail).
Does that make sense?