Simple answer: No truth at all, it's a false assumption with no historical basis.

More in-depth answer: The popular image of a "Mounted Knighted" usually involves a large, fierce warhorse, like a destrier, that has a large size and thick, bulging muscles. But in reality, this "perception" of a horse wasn't reach until the 20th century (in fact, even horses of the Napoleonic Wars of the 19th were contemporarily depicted as more "soft and mild" than "fierce and brutish".)

But in the context of the 13th century, and most of the middles ages, horses weren't ever of a size equal to modern horses (avg.). For instance, both the "Laws and Customs of England" and the "Morgan Bible" depict horses that are quite short and stubby. Contemporary picture for comparisons:

Cavalryman of Alexander III's army: http://upload.wikimedia.org/wikipedia/commons/7/78/Macedonian_Army_Thessalian.jpg

Morgan Bible, 13th century: http://upload.wikimedia.org/wikipedia/commons/a/a6/Morgan-bible-folio-3.jpg

Second Baron's War, 13th century: http://commons.wikimedia.org/wiki/File:Einhard_vita-karoli_13th-cent.jpg

Napoleonic French Horse Guard, 19th century: http://commons.wikimedia.org/wiki/File:Napoleon_Guard_Horse_Grenadier_by_Bellange.jpg

Modern British Horse Guard, 21st century: http://i.dailymail.co.uk/i/pix/2011/04/14/article-1376876-0BA02BF400000578-449_634x400.jpg

edit: formatting

This is false. Horses in the 13th century were much smaller than modern horses. We know this by looking at a few types of artifacts: horse armor and horse transport boats (or recorded schematics for them).

While specific evidence for the 13th century is sparse, manuscripts from the 14th, 15th, and 16th century all show horses that were far smaller than modern horses, and surviving 13th century pieces of armor show horses that are about 14-15 hands high at the withers- which is rather on the small side for a modern field hunter and tiny compared to the average dressage horse or stadium jumper.

Sources-

The Medieval Horse and Its Equipment, C.1150-c.1450, John Clark.

The Warhorse 1250-1600, Ann Hyland

No, they were not larger than modern horses. RHC Davis and Bernard Bachrach both made arguments for larger horses, 18 and 15-16 hands respectively. However, these arguments largely rested on asserting that horses must have been large because armor is heavy and then comparing the sizes of soldiers and horses as depicted in medieval artwork, such as the Bayeux Tapestry. There are some serious issues with this methodology, which become relatively obvious once you start thinking about the roles of verisimilitude and perspective in Medieval art. Or if you take a look at the Bayeux tapestry and compare the size of the human figures to the size of the boats (hint, although they were shallow drafted boats, their sides were more than 2 feet tall).

Newer research has revised these figures down to a maximum of 15 hands. This is based on a number of different sources, including the size of medieval horseshoes (which are pieces of actual physical evidence), consideration of horse breeding techniques, comparisons of horse physiology and anatomy, and consideration of how warhorses were actually used in battle (or not).

I'll summarize each issue quickly. Breeding techniques: We have only bred very large horses over the last 300 years or so (Shire Horses, Belgian Cart Horses), and there's no really giant breed that is the obvious descendant of the supposedly massive warhorse. Instead, large horses tend to be workhorses.

Physiology: Large horses are more adept at pulling things than carrying them directly on their backs. It's also nearly impossible for an average sized human to comfortably ride one, and certainly to fight while mounted on one. Also, the larger the horse the more it needs to eat, making a massive horse massively expensive to maintain and far more likely to be in less than top shape during a long campaign.

In battle: The old, romanticized view of the knight as the medieval tank, riding around with a lance and dominating the field, has been largely rejected. For one thing, the "set piece" battle was relatively rare in medieval warfare -- sieges and raiding were more common. When a cavalry charge did happen, maneuverability and speed were as important as weight and size.

Matthew Bennet sites multiple instances in which either the size of the cavalry was not sufficient to overcome a trained infantry line, or when the size of the horses was actually detrimental to the military effort at hand. He suggests that the largest of horses were not really intended for heavy use in battle, but were actually status symbols.

Sources:

Ann Hyland The Medieval Warhorse from Byzantium to the Crusades

Ann Hyland The Warhorse, 1250-1600

Matthew Bennett "The Medieval Warhorse Reconsidered" in Medieval Knighthood V, Papers from the Sixth Strawberry Hill Conference, 1994

Bernard S. Bachrach "Caballus et Caballarius in Medieval Warfare" in The Study of Chivalry: Resources and Approaches