Arkadaşlarla gayriresmî bir davette daha rahat giyinirim.

arkadaş = “friend” + -lar = plural + -la = comitative case meaning “with.”
Together arkadaş-larla means “with friends.”

gayri- is an Arabic-derived prefix meaning “non-” or “un-,” and resmî means “official.” So gayriresmî = “unofficial” or “informal.”
The circumflex (ˆ) on î signals a historically long vowel or slight palatalization. In modern Turkish you’ll often see it written without the accent (gayriresmi), and the pronunciation difference is minimal in everyday speech.

Turkish locative case (-de/-da) means “in/at/on.” Since davet ends in a consonant, it takes -te (“davet + te → davette”). The doubled t comes from regular consonant assimilation.
davette therefore means “at the event/party/invitation.” In Turkish, davet can refer both to the invitation and the gathering itself.

daha = “more”, used to form comparatives.
rahat = “comfortable.”
So daha rahat = “more comfortable” or “more comfortably.”
If you just want “comfortably” (without comparison), you could also say rahatça giyinirim (“I dress comfortably”), but daha rahat explicitly compares to some other situation.

1. giy- = root meaning “wear/put on.”
2. -in- = middle voice suffix, forming giyin- = “to get dressed.”
3. -ir- = aorist suffix, giving a general/habitual sense.
4. -im = first person singular ending.
   So giyin-ir-im = “I (habitually) get dressed/wear.”
   Here it expresses a regular tendency: “I tend to dress more comfortably…”
   (Note: continuous/progressive would be giyiniyorum, but that would mean “I am dressing right now.”)

• giymek (transitive): “to put clothes on (someone)” or “to wear” when reflexive.
• giyinmek (intransitive/middle): “to get dressed,” “to dress oneself.”
• giydirmek (causative): “to dress someone else.”
In our sentence we need “I dress myself/wear,” so we use giyinirim (from giyinmek).