We just had a spectacular June moon conincident with the summer solstice - see this article on the BBC for example.
Why does the moon (or the sun) look bigger when it's on the horizon? First of all, the image of the moon isn't actually smaller when it's high in the sky. Compare it to a coin held at arm's length when it's high and when it's low and you'll see it's the same. (Though the moon can be bigger on particular days, such as we had recently).
The explanation is that we judge the moon to be futher away when it is on the horizon, because we have objects on the horizon for comparison. See this page at IBM for more details.