奇 is a controversial one.
There are sites that say it is a person on a horse being original form of 骑, but there are no old glyphs with the horse character in the ancient script.
Easiest interpretation would be large 大 and gasping 可 (mouth + breath). A person is gasping in awe at a large peculiar object. 可 can also double as phonetic, although they both don't really sound alike.
怪 is combination of heart 心 and 圣, which is a hand holding a lump of clay. This is probably an ancient form of 块 (which had another old variant form 凷). Most likely just a phonetic component, but you could think of it as an unpleasant feeling like having a lump in your throat or heart.
奇 is more neutral sounding, meaning strange but not necessarily in a bad way. It can even be used positively, being something wonderful or remarkable, like 奇迹 miracle. It is almost never used alone, possibly because it has many other homophones/similar sounding characters.
怪 sounds like ghost 鬼 and is almost always used in a negative tone. For example, monster 怪物, blame 责怪 (责 is often omitted if used in this sense), and a vulgar slang dialectal term 怪𡳞 which means annoying. 怪 can be used alone for blame or strange as it doesn't have as many common homophones.