"さえ" is used to emphasize an example that you usually can do or cannot do. That is to emphasize the different current condition or circumstance from the usual, by listing a particular example.
To take Poisonous Apple-san's examples;
Ex1: お酒に酔いすぎて真っ直ぐに[さえ]あるけない。
Because I am drunk too much I cannot [even] walk properly.
=> You can usually walk straight, but you cannot at the moment under the influence.
Ex2: 今日は気分が良く、空を飛ぶこと[さえ]出来るような気がする。
Today, I feel so good that I feel like I can [even] fly.
=> You cannot usually fly, but you feel so good that you feel like you can. (imaginary example)
Ex3: 2日間なにも食べていなかったので、昨日の残飯で[すら/さえ]おいしく感じる。
[Even] the leftover from yesterday tastes good since I have never eaten anything for 2 days.
=> You usually don't feel the leftover is tasty, but now you do under the given circumstance.
I hope this will help.