Why does the past tense form of sleep have a weak suffix?

Meaning: to sleep is a strong verb in the Germanic languages. While I'm quite aware that strong vs weak anything has very little bearing on modern English, this is still something that puzzles me.

Sleep and slept do show some kind of ablaut, but then where does the -t come from? Why was it added to the end of this verb? I know very little about the specifics of English etymology, and only really have the viewpoint of studying other Germanic languages.

Was it just something as simple as not wanting "slep" as a word and adding the -t made it easier to say/understand? Or just some kind of leveling to make it like wept, kept, etc.?

As John Lawler mentioned in the comments, sleep in present-day English is an irregular weak verb. The synchronic classification of this verb is not related to the conjugation of cognates in other languages, past or present.

Irregular weak verbs show the dental suffix characteristic of weak verbs, but also show other irregular changes; in this case, a vowel change that can be characterized (from certain somewhat abstract perspectives) as "shortening", /iː/ > /ɛ/ ("long e" to "short e"). Wept and kept are other examples of this vowel change.

I don't know enough historical linguistics to explain the diachronic development of the verbs sleep (in English) and schlafen (in German), or why one is weak and the other strong. It is not unprecedented for verbs to move between the strong and weak classes in Germanic (e.g. present-day English help is weak, even though German helfen is strong).

The OED says:

Old English slápan, slǽpan, slépan (past tense slép, slépon, past participle -slápen, etc.), = Old Frisian slêpa (West Frisian sliepe, East Frisian slepe, North Frisian slêp, slîp), Middle Dutch and Dutch slapen, Old Saxon slâpan (Middle Low German and Low German slapen), Old High German slâfan (Middle High German slâfen, German schlafen), Gothic slêpan (past tense saislêp, -zlêp, past participle slêpans); wanting in Scandinavian. Besides the strong conjugation (with reduplicated past tense) Old English also had the weak forms slǽpte, slépte, and after the 14th cent. the strong conjugation disappears from the literary language. A similar change has taken place in West Frisian, where the past tense is now usually slepte, past participle slept. The Middle English slēped (modern Scottish sleepit) may represent the northern Old English forms slépade, plural slépedon (West Saxon infinitive slápian).

It is possible that the weak forms slǽpte, slépte, properly belonged to a causative verb corresponding to Middle High German (ent)-slæfen, older or dialect German schläfen, although no trace of this usage appears in Old English texts. The infinitive of this would have had the form *slǽpan, Mercian and Anglian *slépan, and would consequently have been identical with the infinitive of the strong verb, except where the latter had the special West Saxon form slápan. The strong past tense is frequent in Middle English, and the strong past participle is occasionally found (compare also aslopen adj.); traces of strong conjugation appear in some modern dialects, but it is possible that these are new formations.

I confess that I don't understand what is meant by "the strong conjugation (with reduplicated past tense)" with respect to Old English since I only see reduplication in the Gothic past-tense form, not in any of the other past-tense forms listed in the OED article.

"Slep" is attested, but it doesn't look very much like I'd expect a strong past-tense form to look in modern English, in that I don't think the development of the Old English long vowel /eː/ to Modern English /ɛ/ would be regular in this context. The OED seems to agree; it gives the following list of forms that it classifies as strong past tense:

pa. tense. a. strong. OE–ME slep (OE sclep), OE, ME slæp, sleap, ME sliep, ME sleep, slepp, slepe (18 dial. slape). pl.OE slepon, -un, -an, OE, ME slepen (ME slupen), ME slepe. The common dial. form slep is probably for slept.

I would expect the regular reflex of the Old English strong past tense to be pronounced /sliːp/ in Modern English (perhaps homophony with the present-tense form was part of why it died out).

re the "-t". If we said sleeped instead of slept, the "suffix" would still be pronounced as a t. That has more to do with the voiceless p than the ablaut. In German the vowel change is quite similar - er schläft, er schlief, though certainly not identical, and surprisingly somewhat opposite from the English in this third person example.