How to simplify my Equipment method?

First off, this is not terrible - as it works :)

However, I would do 2 things to come to mind immediatley:

Generalize your CheckForWeapon method. If you replace the hardcoded strings with a variable you could pretty much shorten your code to several lines and it will be much more robust and future proof:

 function CheckForWeapon ( var weaponID)
 {
     if ( hasWeapon )
     {
         if ( equipWeapon.ContainsKey(weaponID) )
         {
             Debug.Log("Found entry " + weaponID + " in inventory!");
             mesh = Instantiate (Resources.Load(weaponID) ) as GameObject;
             mesh.transform.parent = rh.transform;
             mesh.transform.position = rh.transform.position;
         }

Now you should be able to have any number of N weapons in your inventory and not have to treat them differently.

The next thing is more of a design thing, which you might want to leave for later - however, in your example there is a close coupling between the inventory and the player character, this makes it unusable in future scenarios (let's say you want to add an inventory to your enemies for loot, or any other use).

To handle this, break the ties between the inventory and the player entities, as they are two distinct entities which really should not know of one another to operate properly. What it means is the player should be the one equipping and handling items attached to it, the inventory is a rather dumb component only aware of the items it contains and wether or not they exist.

a rough outline of that would be

 Inventory {
     /* Checks if an item exists in the inventory, returns true if does and false otherwise
     bool HasItem( itemID );
 
     /* Checks if an item exists in the inventory, if does it returns a reference to an instantiate copy of the item, otherwise - null. 
     GameObject GetItem( itemID );
 }

 Player {
 
     /* Reference to my inventory */
     Inventory m_inventory;
 
     EquipFromInventory( itemID )
     {
        /* Make sure we have an active inventory */
        if ( m_inventory ) 
        {
              /* Make sure the item exists in the inventory
              if ( m_inventory.HasItem( itemID ) )
              {
                  /* Get the item - we assume inventory handles the instantiation when we request it */
                  GameObject item = m_inventory.GetItem( itemID );
 
                  /* The actual equipping of the item */
                  item.transform.parent = rh.transform;
                  item.transform.position = rh.transform.position;
              }
        }
        else
        {
            Debug.LogWarning(" Trying to access unreferenced inventory! did you lose your satchel? ");
        }
     }
 }

It's put in a C-esque notation, but you should be able to understand the general underlying principles. Now, this is more of a design thing, and there are no absolute rights or wrongs (as you said, your version works), so down the line, whatever it is that works for you - works.

Good luck :)