Stjórnandinn segir að hann muni afgreiða endurgreiðsluna í dag.

In Icelandic the definite article is usually attached to the noun as a suffix.

• stjórnandi = administrator/manager (indefinite)
• stjórnandinn = the administrator/manager (definite, nominative singular)
  The ending also reflects case/number: here it’s the subject, so it’s in nominative singular.

Main clauses in Icelandic follow the V2 pattern: the finite verb typically comes in the second “slot.”
Here the first slot is filled by the subject Stjórnandinn, so the verb segir comes next:

• Stjórnandinn (slot 1) + segir (slot 2) + rest.

að introduces a subordinate (dependent) clause, similar to that in English. It signals that what follows is reported speech/thought/content:

• segir að ... = says that ...

After að, you’re in a subordinate clause, and Icelandic does not use V2 there in the same way. The finite verb typically comes after the subject:

• að hann muni ... (subject hann before finite verb muni)
  Whereas in a main clause you might see V2-like order more often.

It’s a form of the verb munu used to express futurity (often like will). In this sentence muni is commonly analyzed as present subjunctive, which is frequent in reported/embedded clauses after verbs like segja.
In a direct main-clause statement you often see:

• Hann mun afgreiða ... (more “straight” future in a main clause)
  But after segir að, muni is very common/idiomatic.

Because it’s the main verb controlled by munu. The structure is:

• finite auxiliary-like verb muni
  • infinitive afgreiða
    Comparable to English will + verb.

endurgreiðsluna is accusative singular definite (feminine), used as the direct object of afgreiða.

• endurgreiðsla (nom. sg., indefinite)
• endurgreiðslu (acc. sg., indefinite)
• endurgreiðsluna (acc. sg., definite) = -na attached to the accusative form.

You often learn noun gender with the dictionary form, but there are clues:

• Many nouns ending in -a are feminine (though not all).
  Also, the definite accusative ending -una is a strong signal you’re dealing with a feminine noun in accusative singular definite.

A natural direct version would be something like:

• Stjórnandinn segir: Hann mun afgreiða endurgreiðsluna í dag.
  or simply:
• Hann mun afgreiða endurgreiðsluna í dag.
  Notice mun (main-clause form) is typical there, instead of embedded muni.

A few common ones for English speakers:

• Stjórnandinn: the tj sound is like a palatalized t (not English sh). Stress is on the first syllable: STJÓR-.
• segir: the g is soft here (often like a voiced/approximant sound, not a hard English g).
• afgreiða: ð is typically like the th in this (voiced).
• endurgreiðsluna: long compound; keep primary stress near the start (EN-), and pronounce ei like the Icelandic diphthong (not English ee).