"The short answer is that they are not always weaker in fact. Some ionic compounds have very strong bonds, while some covalent bonds are quite weak. Usually however, it is easier to break an ionic bond than a covalent one. What determines the actual strength of a bond is quite complex, but let me try to explain the basic principles.
Bonding has everything to do with electrons. In ionic bonds, an element loses an electron to another element. The two elements are then bonded by a force of attraction based on electrostatics, or Coulombic force; it is due to the fact that a positive charge and a negative charge attract each other (like the two opposite ends of a magnet). Since one element has a negative charge and the other has a positive charge they are held together in a bond. I will give an example. Sodium chloride, NaCl, or more commonly known as table salt, is an ionically bonded compound. The sodium, Na, will lose one electron (to form a Na+ ion), and the chlorine, Cl, will gain that electron (to form a Cl- ion). Since an electron has a negative charge, the sodium has a positive charge (lost a negative), and the chlorine has a negative charge (gained a negative). Just like north and south magnets, the two are attracted to each other and are bonded. The bond strength is determined by how much charge each element has. For instance, you can also have ions that have more than one charge. Common ions like this are Mg2+, Fe3+, and O2-. Because each ion has more than one charge, the attraction is even stronger because the Coulombic force is directly related to the size of the charge on each ion.
In covalent bonds, elements share electrons. No electrons are given away or taken completely, but are instead shared between the elements. For example, Nitrogen gas, or N2 is simply two nitrogen atoms bonded together. There is no reason for one N to give an electron to the other one. They both want to hold on to their electrons exactly the same amount! (As opposed to NaCl in the example above, where Cl needs an extra electron, and Na needs to get rid of one to become most stable). The strength of a covalent bond is determined by a complicated set of properties of the atoms involved. What makes a covalent bond weak or strong is really beyond the scope of this discussion. (Briefly, the strength of covalent bonds is related to the nature of the atomic orbitals involved in the bond, specifically the overlay of the orbitals involved in the bond, and the size of the energy difference between the orbitals. If that doesn't make sense, don't worry too much about it... it's complicated stuff!)
In general, ionic bonds are easier to break, but it depends on several factors. For instance, breaking ionic bonds is very easy to do when you dissolve an ionic compound in water. Water is special because it effectively reduces this Coulombic attraction between the plus and minus charges on the ions (it kind of acts like a barrier or shield, blocking the plus from seeing the minus). However, if you were trying to break an ionic bond in a perfect vacuum, it would be much harder (because without something to block the attractive force, it is quite strong). Another thing to consider is that ionic compounds usually form 3-dimensional crystals, where there are many many ionic bonds, whereas covalent bonds tend to be found in isolated molecules (except for network covalent compounds like diamond or silicon). The fact that ionic bonds form a 3-D structure in a crystal makes a huge difference. So part of what makes this question so complicated is that you kind of comparing apples and oranges."
