Have digged and drunk and dried up - The meaning seems to be - "Mountains do not stop me - I cross them even in my chariots. Deserts do not stop me - I dig wells there, and drink the water. Rivers do not stop me - I pass them as easily as if they were dry land."
The rivers of besieged places - Rather, "the rivers of Egypt." The singular form, Mazor (compare the modern Misr and the Assyrian Muzr), is here used instead of the ordinary dual form, Mizraim, perhaps because "Lower Egypt" only is intended. This was so cut up with canals and branches of the Nile, natural and artificial, that it was regarded as impassable for chariots and horses. Sennacherib, however, thought that these many streams would prove no impediments to him; he would advance as fast as if they were "dried up."

I have dipped and drunk strange waters - I have conquered strange countries, in which I have digged wells for my army; or, I have gained the wealth of strange countries.
With the sole of my feet - My infantry have been so numerous that they alone have been sufficient to drink up the rivers of the places I have besieged.

2-Kings 19:24 refers to the intended conquest of Egypt. Just as Lebanon could not stop the expeditions of the Assyrians, or keep them back from the conquest of the land of Canaan, so the desert of et Tih, which separated Egypt from Asia, notwithstanding its want of water (cf. Herod. iii. 5; Rob. Pal. i. p. 262), was no hindrance to him, which could prevent his forcing his way through it and laying Egypt waste. The digging of water is, of course, not merely "a reopening of the wells that had been choked with rubbish, and the cisterns that had been covered up before the approaching enemy" (Thenius), but the digging of wells in the waterless desert. זרים מים, strange water, is not merely water belonging to others, but water not belonging to this soil (Drechsler), i.e., water supplied by a region which had none at other times. By the perfects the thing is represented as already done, as exposed to no doubt whatever; we must bear in mind, however, that the desert of et Tih is not expressly named, but the expression is couched in such general terms, that we may also assume that it includes what the Assyrian had really effected in his expeditions through similar regions. The drying up of the rivers with the soles of the feet is a hyperbolical expression denoting the omnipotence with which the Assyrian rules over the earth. Just as he digs water in the desert where no water is to be had, so does he annihilate it where mighty rivers exist.
(Note: Compare the similar boasting of Alarich, already quoted by earlier commentators, in Claudian, de bello Geth. v. 526ff.:
cum cesserit omnis
Obsequiis natura meis? subsidere nostris
Sub pedibus montes, arescere vidimus amnes.
v. 532. Fregi Alpes. galeis Padum victricibus hausi.)
יאורי are the arms and canals of the Yeor, i.e., of the Nile. מצור, a rhetorical epithet for Egypt, used not only here, but also in Isaiah 19:6 and Micah 7:12.

Strange waters - Such as were never discovered by others. Dried up - And as I can furnish my army with water digged out of the earth; so I can deprive my enemies of their water, and can dry up their rivers, and that with the sole of my feet; with the march of my vast and numerous army, who will easily do this, either by marching through them, and each carrying away part with them: or by making new channels, and driving the waters of the river into them.