I've gotten better at listening to native speakers.
Thanks to my lessons on DMM, I can now understand native English speakers.
おかげで= Thanks to
DMMのレッスン= DMM's lessons-->my lessons on DMM
聞き取れるようになった」= I've come to grasp by listening --> I can now understand
ネイティブの英語= native English speaker's English-->native English speakers.
何かのスキルが良くなってきた場合、このような表現は便利です:
...has improved, ...has gotten better, I can...now,
例えば絵を描くことを上達になったら、"my drawing skills have improved," "my drawing ability has gotten better/I've gotten better at drawing" or "I can draw much better now" と言います。
Thanks to the DMM lessons, I am now able to understand what the native speakers of English are saying.
The DMM lessons have helped me to understand English spoken by its native speakers.
ご質問ありがとうございます。
① "Thanks to the DMM lessons,"＝「DMMレッスンのおかげで」
"I am now able to understand what the native speakers of English are saying."=「今は、英語のネイティブスピーカーが言っていることが理解できる。」
② "The DMM lessons have helped me to understand English spoken by its native speakers."＝「DMMレッスンは、ネイティブスピーカーが話す英語が分かるように手助けしてくれた。」
ご参考までに。
アメリカ合衆国 
