I have read the explanation for this in several textbooks, but I am struggling to understand it via Archimedes' principle. If someone can clarify with a diagram or something so I can understand or a clear equation explanation that would be great. 
Good question.
Assume we have one cube of ice in a glass of water.  The ice displaces some of that water, raising the height of the water by an amount we will call $h$.
Archimedes' principle states that the weight of water displaced will equal the upward buoyancy force provided by that water.  In this case,
$$\text{Weight of water displaced} = m_\text{water displaced}g = \rho Vg = \rho Ahg$$
where $V$ is volume of water displaced, $\rho$ is density of water, $A$ is the area of the ice cube base and $g$ is acceleration due to gravity.
Therefore the upward buoyancy force acting on the ice is $\rho Ahg$.
Now the downward weight of ice is $m_\text{ice}g$.
Now because the ice is neither sinking nor floating, these must balance.  That is:
$$\rho Ahg = m_\text{ice}g$$
Therefore,
$$h = \frac{m_\text{ice}}{\rho A}$$
Now when the ice melts, this height difference due to buoyancy goes to 0.  But now an additional mass $m_\text{ice}$ of water has been added to the cup in the form of water.  Since mass is conserved, the mass of ice that has melted has been turned into an equivalent mass of water.
The volume of such water added to the cup is thus:
$$V = \frac{m_\text{ice}}{\rho}$$
and therefore,
$$Ah = \frac{m_\text{ice}}{\rho}$$
So,
$$h = \frac{m_\text{ice}}{\rho A}$$
That is, the height the water has increased due to the melted ice is exactly the same as the height increase due to buoyancy before the ice had melted.

Edit: For completion, since it is raised as a question in the comments
Melting icebergs boost sea level rise, because the water they contain is not salty.

Although most of the contributions to sea-level rise come from water and ice moving from land into the ocean, it turns out that the melting of floating ice causes a small amount of sea-level rise, too.
Fresh water, of which icebergs are made, is less dense than salty sea water. So while the amount of sea water displaced by the iceberg is equal to its weight, the melted fresh water will take up a slightly larger volume than the displaced salt water. This results in a small increase in the water level.
Globally, it doesn’t sound like much – just 0.049 millimetres per year – but if all the sea ice currently bobbing on the oceans were to melt, it could raise sea level by 4 to 6 centimeters.

