Is metathesis initially just a speech error?And,if so,how does a metathesised word become the standard form of that word?I can't understand how such a shift could happen,because it is so different from other forms of linguistic changes.
There is no one answer, because there are multiple types of metathesis and therefore multiple causes. My understanding of the history of Straits Salishan metathesis (Saanich, Klallam, Lummi and so on: example mtəqʷ (non-actual aspect) ~ mətqʷ (actual aspect) "put in water") is that it is the result of differential stress assignment depending on aspect, and syncope of the unstressed vowel. Some Palestinian Arabic dialects similar have (apparent) metathesis exemplified by yiktib 'he writes', yikitbu 'they write', which again reflect an interaction between syncope, epenthesis and vowel harmony (as discussed by Kenstowicz in "Vowel Harmony and Metathesis in Palestinian Arabic").
Maltese (Arabic) likewise has a metathesis process affecting /CV-CRVC-V/ imperfectives where R is any sonorant, see (ni-tlob ~ ni-tolb-u 'I/we pray' contrasted with nikteb ~ ni-ktb-u 'I/we write'). In this case, a regular rule would otherwise delete o in underlying /ni-tlob-u/ resulting in *nitlbu (a sonorant trapped between consonants). The exact sound change path for this example of metathesis is lost in history, but probably involves an interaction between vowel harmony (which exists in Maltese anyhow) and reduction, where a demi-vowel is phonetically inserted before the sonorant in intermediate nitV̆lŏbu (where ŏ is extremely reduced, and coarticulatory harmony then turns indistinct V̆ into phonological [o]). The underlying premise of this kind of analysis for VC metathesis is that since the features of vowels and consonants are typically disjoint, there can be intermediate states with reductions in vowels gestures, and insertions of generic vocalic gestures (releases), where phonetic coarticulation is responsible for giving the inserted vowel the features that the "doomed" vowel has. For this kind of metathesis, segment movement is not a renumbering of segments, it is a barely perceptible and historically continuous shifting of vowel place gestures, which overlap consonants to varying degrees.
There is another kind of metathesis involving sibilants in clusters, exemplified by Faroese fɛsk-ʊr ~ fɛks-t 'fresh (masc, neut)', where /fɛskt/ → [fɛkst]. The above barely-perceptible gesture shifting scenario doesn't seem plausible in this kind of change. The consequence of metathesis is that k is in a perceptually somewhat better position (after a vowel, where it can be identified by V-to-C formant transitions), and s doesn't suffer any (sibilants between consonants are not hard to identify.
There are examples of non-systematic metathesis in language change, for example where 'ask' comes out as [æks], but 'mask' does not come out as *[mæks]. Such examples might originally have been errors in the sense of being misperceptions of standard pronunciations. This paper discusses child and adult language acquisition and metathesis, such as Japanese children metathesizing nezumi → [nemuzi]. I think a better understanding of patterns of metathesis across languages in child language acquisition would help to understand the basis for metathesis as a sound change.
Not just a speech error. Depending on the language, it brings grammatical/semantic changes too, checkout the language -> Saanich's use of metathesis. i.e for some language it might bring a change in the meaning of the word or it might change the gender, number , tense , aspect etc.
Checkout this wiki page's examples for starters -> https://en.m.wikipedia.org/wiki/Metathesis_(linguistics)
Focusing more on the speech error side, if the error makes it easier to pronounce or distinguish from some other word or has become more popular (because of a hashtag maybe! ;) ) in the same language then the changed version becomes more and more acceptable, it's actually like any other change Linguistics, 'for the ease of its native speakers'.
Hope this is some food for thought and helps you...