Ben resmi masadan çıkardım.

“Ben” is the 1st-person singular subject pronoun (“I”). In Turkish, the personal ending on the verb (here –dım in çıkardım) already shows that the subject is “I,” so you can omit Ben in everyday speech:
• Resmi masadan çıkardım.
Speakers include Ben mainly for emphasis or contrast (“I, personally, took the picture off the table”).

The –i is the accusative case suffix marking a definite direct object. In Turkish, if the object is specific or known (here the picture), you add -ı/-i/-u/-ü according to vowel harmony.
• resim (picture) + i → resmi (“the picture”)
If it were indefinite (“a picture”), you’d say bir resim and omit the accusative suffix.

Masa means “table.” The suffix –dan is the ablative case ending, which means “from” or “off.” Vowel/consonant harmony dictates masa + dan → masadan:
• masadan → “from the table” or “off the table”

• çıkmak (intransitive) = “to go out,” “to exit”
• çıkarmak (causative/transitive) = “to cause to go out,” i.e. “to remove/take something out”
Since you’re removing the picture, you need the transitive form çıkarmak.

• root: çık- (“to go out/exit”)
• causative suffix: –ar → çıkar- (“to make go out” = remove)
• definite past tense suffix: –dı → çıkardı- (“removed”)
• 1st-person singular suffix: –m → çıkardım (“I removed”)

Turkish typically follows a Subject-Object-Verb (SOV) order. Elements like place or time phrases come before the verb. In your example:
[Subject] Ben [Object] resmi [Place] masadan [Verb] çıkardım

Use bir + noun and drop the accusative suffix:
“Ben masadan bir resim çıkardım.”
Here bir resim = “a picture,” and because it’s indefinite, you do not use –i.