Is there any logic to this assertion?

Yes there is definitely logic there.

Marginal revenue is the additional amount of money a firm gets for selling one more unit of their product. Marginal cost is the additional amount of money they need to spend to make that unit (i.e. by hiring another worker or buying more supplies etc). If your marginal revenue is higher than your marginal cost you are making money for that additional unit. So you would want to keep making more so you can make more money. If your marginal cost is higher than your marginal revenue then you are losing money by selling that additional unit which a firm wouldn't do.

If your marginal revenue equals your marginal cost then you've hit your break even point - you can't make more money by producing more and you lose money by producing more. So you've maximized your profit.

There are some assumptions with this. One is that the MR and MC curves do meet (if you always get $7 revenue and it costs $5 to make you maximize profit by making an infinite amount). And that before they meet MR is higher than MC and after they meet MC is higher than MR.

If you want to try drawing some generic MC and MR curves and then figure out where the profit margin is - you'll see quite quickly how this statement makes sense.