I'm seeing several answers that seem to be tiptoeing around the problem but not quite hitting it head-on...
If you have time, have a look at this Serious Eats article, The Food Lab: Real Ice Cream Without an Ice Cream Machine. Although it's about making ice cream without a machine, many of the principles apply to machines as well, because the problem you're experiencing is the same as those experienced by those without ice-cream makers: Ice crystal formation.
Contrary to what Chef's answer says, you do not need to keep ice cream at above-freezing temperatures in order to prevent crystals. As Mollie has correctly pointed out, that is precisely the purpose of overrun; the airier the ice cream, the less opportunity there is for ice crystals to form. But that's not the whole story. You don't just need to incorporate air, you need to keep it dispersed so that that the liquid never gets too dense in one place.
Ice-cream makers are supposed to do both of these things. By churning the cream while it freezes, it keeps the mixture dispersed and introduces more air (overrun).
It sounds to me like what's happening is that your ice-cream maker is simply not getting cold enough. If it actually brought the temperature of the ice cream to zero or below, then the consistency would not change after putting it into an actual freezer. So you're getting not-quite-freezing ice cream out of the ice-cream maker which is great, but then you're putting it in a freezer where that not-quite-frozen liquid is forming large ice crystals as it freezes for real.
So your solutions to this problem are:
Invest in a higher-end ice cream maker that chills better (this may or may not be practical).
Introduce extra overrun by pre-whipping the cream before putting it into the machine. Whip it until soft peaks form, then fold it into the mixture (don't stir!). You can do the same with the egg whites as well. This may give you an undesirable amount of overrun, like the 94% that's in Breyer's, but it will make it almost impossible for crystals to form.
Flash-freeze it with dry ice or liquid nitrogen. This may not be practical either but is the surest way to prevent ice crystals from forming, since the main thing crystal formation requires is time.
If you can't flash-freeze, then try packing the churned ice cream into ice trays as Kenji (Serious Eats) does with the "raw" ice cream, then give it a whirl in a food processor once it's frozen.
Finally, one thing that even Serious Eats neglected to mention is to use a stabilizer. Xanthan gum is great for ice cream and will help keep it at "whipped" consistency while it's in the freezer. You don't need a lot - use about 0.5% of the total weight at most. Stabilizers in particular are what will allow you to freeze the ice cream for longer, and prevent that 1-2 day "expiration date" that was mentioned by an earlier answer.
Any or all of these things will improve the smoothness of the ice cream and help it to survive the freezing process.