A typical one of these MOSFETs will switch 40A with 5v Vgs. But notice that the typical threshold voltage is 3.1 but max is 3.9 (difference of 0.8V).
If you apply 5V to a MOSFET with a high threshold, it won't be typical any more, it will work like the (5.0 - 0.8 ) = 4.2V curve (which they don't show but you can bet it's worse than the lousy 4.6V curve that they give you).
If you really have 5V, that is. If your 5V supply is 4.75V (5% lower than 5V) and your driver takes 0.2V you are even worse off.
When the MOSFET is 'saturated' its current capability is governed by Vgs (gate voltage). Below the critical current level, it's called 'linear' and follows the Rds lines. Once it is carrying all the current that the channel will allow, the curve departs from the Rds lines and moves to the saturated curves, which are like current sources, and are pretty much independent of Vds. So Vds rises. If you multiply Vds by the current, you'll notice you can't do this for very long since it is outside the power ratings.
MOSFETs can be confusing since the 'saturated' and 'linear' sections of operation are in opposite places when you compare them to a BJT curve.