Aside from the fact that Bottom is a fine example of being down-to-earth, as he astutely notes 'Yet to say the truth, reason and love keep little company now-a-days', giving fresh insight and perspective on love, how does his character or particular actions make him stand out as 'the only real character in the play'?
The title of the play indicates that very little is real. Bottom and the other mechanicals who compose the subplot are, for the most part, satirized as representative of certain types of people. As such, then, Bottom is more real since he represents more than just himself.
What an intriguing essay question to think about! I guess you have a series of points to think through in answering this question, and you will need to analyse carefully Bottom´s involvement and contribution to the play as a whole by looking at his speeches and his involvement.
One place to start is by situating Bottom very firmly in the "real" world - he is part of the Mechanicals of Athens who are preparing a play for the wedding of Thesus and Hippolyta. However, he is the one character in the play that crosses the line between the real world and the fairyworld - he plays a significant role in both worlds, as Titania´s "lover" with his ass´ head. You will also want to carefully consider his speech when he wakes up after his "dream", which is Bottom´s dream, "for it has no Bottom". Of course, above all, you will want to carefully define "real" in your response. How is Bottom different to the other characters? How is he similar? Hopefully this will give you some points to get you started.
That's an interesting question, and, of course, it all depends on what is meant by "real". If meaning "round", I think it could be argued that none of the characters, with the possible exception of the young lovers, are truly round in the way we think of characters in plays and novels today. We don't know all of the motivations or understand the actions of any of the characters. There are too many characters, in the medium of a play of limited length, for that to happen.
Real -- meaning down-to-earth, as poster 1 says, can be argued for Bottom. But all of the Rude Mechanicals are trying to put on airs by putting on this play, and all of them are out of their element. They are certainly humble and straightforward, but all of them, including Bottom, are figures of fun, and are meant, mostly, to appear silly. There are some moments of real pathos in Bottom's part, but, as with most of the lower-class characters in Shakespeare's play, their suffering is played for laughs, rather than for emotion. That we may see it as more touching today is just Shakespeare's good writing, not necessarily Shakespeare's intention, or how his audience received it. But this play, as most of Shakespeare's plays, is wonderfully flexible.
If I were answering this question, I would say that the only characters that are real to me (meaning that I understand them, their motivations, and their actions, and can, in certain ways, identify with them) are Titania and Oberon, and, to a lesser extent, Puck. They are having a marital disagreement, and while the resolution is not necessarily what I would have chosen in their situation, it does reflect a certain amount of marital compromise. The silliness of love-potions and fairies doesn't detract from their reality to me, as characters rather than supernatural beings. I find the young lovers more difficult to understand, but other people identify with them very easily. I find the Rude Mechanicals, including Bottom, closest to caricatures, and the furthest from real. But this is entirely an opinion - many arguments could be made from other perspectives.
I don't agree that he is the only real character in the play, but in terms of your quote, he is the only character of any substance who is not in love or being pursued by someone in love with him--that is, until he is the victim of Titania's love potion and her very lavish attentions. He is an idiot, lacking reason, but he also lacks love.