How to save money in Germany

First and foremost: Germany is not cheap!
The official survival minimum (poverty line) of €8700/year (€725/month) per person does NOT allow a comfortable life - and in big cities you need a lot more.
That's why (for your own financial wellbeing) you cannot get a visa if you have less.
Please note that this is cash in your pocket, after tax and social security deductions (which are high). For example, if you live in Germany with spouse and two kids and you earn €4500/month before tax, which sounds a lot, you get €3000/month after tax and thus are in fact barely above the poverty line for four. In this case, you will only save if you forgo almost all luxuries (believe me: there are plenty of temptations to spend lots).
Especially costly items in Germany are restaurant meals, accommodation, travel, exotic foods (which might include what you are used to from home), cigarettes.
Food and daily necessities are comparatively cheap, especially if you buy at discount stores. Public transport and cars are also affordable.
Schooling is free, child care and universities heavily subsidized.

Beppi has provided a concise and well thought out answer. To expound on it there is a lot of competition from chain stores for food, cloths, house hold items, electronics  etc. One can go to Ikea for good furniture. Or look online for further deals. So most necessities are relatively cheap. Once one wants brand names then things get more expensive.  Eating out cost more than cooking oneself but it is also relative. One can often find a simple, cheap Turkish or Asian restaurant where they can get a good meal for say 6 to 8 Euros – or one can go to a fancy restaurant and spend multiple times as much for basically the same thing.

For people on a super limited budget there are websites where one can simply ask for things and people donate them. Or there is what is called Sperrmüll where people throw out things that don't fit in the normal garbage containers like furniture, household items etc. that they no longer want and it gets picked up and trashed. Of course lots of broken worthless things go out with the Sperrmüll but also lots of perfectly good things. The reason is that especially in the cities, people tend to have limited space for storage and they haven't the time or energy to sell or even donate things to charity organizations. Thus, they just throw out perfectly usable things. Want some skis? No problem, people buy fancy new ones and throw out the old ones even though they are perfectly good; furniture, dishes etc. are abundant. I personally would not want to take someone's old mattress or used underwear but nothing wrong with taking that retro lamp or ceramic flower pot or real wood cabinet.

There are also many flea markets. There are some regular dealers that sell new things for relatively expensive and others that sell things for very cheap, chances being they took a lot of the stuff from Sperrmüll themselves. Weekly flea markets tend to have more expensive dealers while yearly or occasional ones often attract people just trying to clear out their unneeded junk and wanting to get rid of their stuff for whatever price they can get.

The biggest cost factor for most people is going to be accommodation. Finding a good place for an affordable price gets more difficult all of the time and one simply needs some luck. There is always a tradeoff. One might find a relatively cheap place but then what condition is it in? Is it by a noisy street and far away from public transportation, schools and shopping opportunities? One's personal situation will determine which factors are important. If one works at home, has no kids and has a car then they might find a bargain in a small village or outskirts of a city where others would hesitate to live.

Personally, I bike and walk almost everywhere, which saves me around 100 euros per month.  I tend not to shop at discounters, but instead look at the weekly deals of my local supermarket/butcher and plan my meals accordingly.  There are certain items (like Irish butter) that I stock up on when they go on sale.  I also compare prices with the farmers market, because oftentimes certain items are cheaper and fresher there (especially spices).  I tend to limit myself on prepackaged junk food which is overpriced anyway. 

For non food items, I check out ebay and the Kleinanzeigen, or even ask around if anyone is selling whatever it is I need.

Healthcare is super expensive in Germany, compared to other European countries. So if you can, either keep your healthcare provider from back home, or get a job-contract (not a 400 euros contract though, with such a contract you will still have to pay the required minimum of 300euros a month as "selbständig")

The above post is inaccurate: Everyone residing in Germany HAS to have German health insurance, by law - so it is not possible to keep your home health provider (except for short visits or members of foreign military and diplomatic corps stationed here, for which special rules apply).
But it is correct that a change occurs once you earn above €450/month, which makes health insurance cheaper in many (but not all) cases.