Why are the seeds on the outside of a strawberry?

What appear to be the seeds on the surface of the strawberry
are actually multiple tiny fruits.

[For more fascinating adaptations of fruits and seeds, see chapter 7 of my book, Plant life - a brief history]

I should begin by putting both “seeds” and “fruit” in quotation marks, because all is not what it seems when it comes to the strawberry.    The short answer to the question is that the seeds are right where they belong – inside the fruits!

The long answer requires a short lesson in flower structure.  At the most basic level, flowers consist of four series of organs: sepals, petals, stamens, and carpels (sometimes also called pistils).   Sepals are typically green and surround the rest of the flower in bud, and the petals are typically the most conspicuous, often colorful and/or fragrant, parts of the flower inside of the sepals.   Stamens bear the pollen, which will be carried away by wind or animals to another flower, where they will release sperm cells.  The eggs awaiting those sperm cells are located inside embryonic seeds called ovules, and the ovules in turn are inside the carpels.  The hollow part of the carpel that contains the ovules is referred to as the ovary (the more accurate term, “ovulary,” never caught on).

The flower of a strawberry. "r" points to the receptacle, which will swell to become the edible part.  The small ovoid structures on the surface of the receptacle are the carpels that contain one embryonic seed each, and will become the hard-walled true fruits as the seeds matures.  Illustration from the classic botany text by Hill, et al.

In some flowers, including the most ancient, flowers include a number of separate carpels.  This true in many members of the Rose Family (Rosaceae), including strawberries (genus Fragaria) and blackberries (genus Rubus).  The carpels occupy the center of the flower and are mounted on a rounded platform called the receptacle.  Each of the carpels in these plants contains just one embryonic seed.    Among flowering plants in general, there are many variations on carpel structure.  In most, carpels are fused together into a compound pistil, and the number of seeds produced within can number from just one to hundreds of thousands. 

The word “fruit” is used quite differently by botanists and consumers of said objects.  What the botanist calls a fruit is technically the tissues of the carpel/pistil/ovary that expand as the seeds within them mature.  What we call fruits in the grocery store and cookbooks are generally plant parts emerging from flowers that are juicy and sweet, but not necessarily part of the carpels.    Sometimes, these two definitions coincide, but sometimes not.  We tend to think of tomatoes, egg plants, bell peppers, okra, and green beans as vegetables, but they are technically all fruits because they contain the seeds.  Kernels of maize, whole oats, and grains of whole wheat also fruits (with thin, hard ovary walls), while the juicy parts of strawberries and apples that we consume are not technically or entirely part of the fruit.

In blackberries, the fruit is an
aggregate of many simple fruits
mounted on a common receptacle,
as in a strawberry, but the individual
carpels become fleshy rather than the
receptacle.

The flowers of strawberries and blackberries are quite similar, but how they develop into fruits is quite different.   In strawberries, the receptacle expands into the tasty succulent tissue that we crave, while the tiny dry fruits, which each contain one seed, sit on the expanding surface.    In the blackberries, the carpels themselves expand into small, juicy, seed-containing globules, and so are true fruits. 

An apricot, like a plum, cherry, or peach,
 develops from a single carpel.

Apples (genus Malus) and plums (genus Prunus) are also in the Rose Family, and their flowers are similar to those of strawberries and blackberries.  Apples are different in that tissues from the receptacle expand on the outside of the carpels to become the edible portion. If you bite too deeply into the apple (or pear) you encounter the hard inner wall of the true fruit and the seeds within.   The plum (or apricot, cherry, or peach) is more like the blackberry, except that there is only one carpel in each flower.  The carpel wall expands to form the edible tissue, and so is a true fruit.