It is important here to recognise the difference between debt and deficit.
Deficit is the difference between the amount of revenue a government receives (e.g. from taxation) and the amount it spends, any shortfall needs to be made up by borrowing.
Debt is the total amount of money that is owed, not only does this eventually need to be paid back but also has the ongoing cost of interest payments.
So (in simple terms) before you can even begin to pay off the debt you need to have more money coming in than you spend. In practice it is slightly more complex than this as inflation can reduce the effective size of a debt and you can borrow money to pay off debts to get better interest rates, and for a whole country the value of the currency has a significant effect,
To reduce the deficit governments can either reduce spending or increase revenue from taxation. Taxation revenue depends both on the levels of taxation and the total value of the economy compared to what is being taxed e.g. if companies make more profits and employ more people then corporation tax, income tax and national insurance receipts go up. You can increase taxation but there is a danger that if taxes are too high you slow down the economy as businesses either become non-viable or cut costs. Equally VAT revenue depends on consumer spending and business rates are ties to the value of occupied business premises.
At this point is is also worth noting that any business which sells VAT liable products, employs people or occupies premises is directly contributing to government revenue through income tax, NI, VAT and business rates, regardless of the amount of corporation tax it pays.
So essential thinking behind austerity is that you cut spending now to reduce (or at least control the rate of increase of) total debt and the associated interest payments. This means you will have more money to play with in the future while at the same time keeping taxation relatively low. Equally it is not trivial to calculate exactly where the ultimate burden of taxation falls.
The counter argument is that governments are generally able to borrow money at a fairly affordable rate so it is better for them to spend more in the short term to boost the economy and make it up later from increased economic growth and productivity. This can come both from the direct results of more cash going into the system e.g. if the government builds a whole load of roads then that cash goes to contractors and their suppliers who employ people who then spend their wages. In the longer term well targeted spending on health, education and infrastructure will have long terms economic benefits in terms of productivity which will make existing businesses more effective and attract new investment.
What a government can't do though is pump money into the economy indefinitely without getting any sort of return. Ultimately a government can only skim off some percentage of the wealth generated by the economy as a whole. While they very much can borrow or indeed just print more money it ultimately needs to come from somewhere.
The big difficulty is that all of these effects are very hard to measure and there are reasonably logical arguments on both sides. But there is no entirely subjective measure of what approach is 'right' regardless of what criteria you apply.
On one hand neither cutting costs at the expense of productivity nor throwing good money after bad without getting any return are sustainable strategies.
Of course part of the issue is that, by their nature politically motivated governments (and indeed the people who elect them) of any stripe tend to be very bad at long term decision making.
The reality is that it is not a question of austerity vs generosity as that would be a gross over simplification but rather what is the most effective way for a government to spend the resources it has available and this comes down to complex fine details rather than slogans or internet memes.
Ultimately is is not even necessarily about morality or class as it is generally the poorest and most vulnerable who really get it in the neck if an economy implodes...just ask the Greeks.
I would also add that this is written from a fairly politically neutral viewpoint and if anything I would say that both of the main UK political parties are more or less equally poor on this issue.
There is also an effect that public spending in general can boost the economy in the short term by employing people but unless it increases productivity it can be unsustainable. Again this very much depends on exactly how money is spent in very very general terms spending on infrastructure and research and development grows the economy while spending on services is more likely to disappear.