It was known as early as 129 AD that blood was pumped from the heart, this was realized by Galen who saw the beating heart of ripped open gladiator. He experimented on animals and concluded that blood was created in the liver then pumped from the heart to other parts of the body to be consumed. It wasn't until 1628 that William Harvey realized that the intricate blood vessels and construction of the heart and its motion that it was actually circulating throughout the body.

So to answer your question, it was believed that consumed food transformed into blood in the liver then that blood was pumped from the heart to tissue to be consumed, and that bleeding was the equivalent of starving.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3721262/ http://vascularcell.biomedcentral.com/articles/10.1186/2040-2384-1-3

It has long been understood that blood was vital to life, and that the heart pumped blood. For example, Galen wrote that blood originated in the liver and then was pumped through the heart to the lungs and the rest of the body. In Deuteronomy blood is referred to "life" and clearly understood as important.

Up until Harvey (and for some time afterwards) most seem to have thought that blood was synthesized in the body rather than recirculated. This isn't really surprising, given that while it's easy to see the connection between veins and arteries and the heart, there's no obvious connection between arteries and veins. They connect via capillaries, which are microscopic. So there is no visible "loop" for blood to circulate through unless you have a microscope (and even then capillaries are hard to find). There is a very large vein leading from the liver, which no doubt is why Galen came up with his ideas.

Here's a write-up that discusses Galen, Harvey, and some of the others in between them

http://membercentral.aaas.org/blogs/scientia/circulatory-system-galen-harvey

The Chinese knew about blood circulation fairly early on. This was paired with circulation of nutrients and 'qi,' though, so there is a fair bit of mysticism and numerology alongside more 'scientific' stuff.

http://sesp.northwestern.edu/docs/publications/6074956944509ac426aaa6.pdf

Prior to Galen, the ancient Egyptian "scientists" believed that your body was made up of very many channels, through which blood, air and water traveled from your heart to your extremities. They also extrapolated, from watching the irrigation of the Nile, that when a channel was blocked (drought of the irrigated section of Nile) that the surrounding area of your body began to die, much like the farming land of the Nile. They believed that the heart was the main organ responsible for thought, and they connected the heart with the pulse.

Source: Edwin Smith Papyrus

I'm re-reading Frankenstein right now. The answers here provide great context for the times. Thanks for asking this one!