自分のArduino IDEに、SoftwareSerialかNewSoftSerialのどっちが入っているかはこのページの’Program the board’を見ながらチェック。

BoardでATtiny45 (8 MHz)などを選んでBurn Bootloaderをすると、
(もしくは、レゾネータやクリスタルを使ってexternal 20MHzで動かしてもたぶん大丈夫っぽいです)

詳しくはこちらの’Configuring the ATtiny to run at 8 MHz (for SoftwareSerial support)’を。

You cannot use Arduino’s Serial library for ATtiny 45, 85, 44 and 84 (and any tiny series, I guess).
So you can use SoftwareSerial library or NewSoftSerial library instead of it.

(Check which you have on your Arduino IDE, SoftwareSerial or NewSoftSerial with seeing ‘Program the board’ on this page.

But, after programming the board, it didn’t work.
Then after struggling for a while, I found I forgot to burn bootloader on the board.

ATtiny series run at 1MHz by defalt,
so you need to configure the board to run at 8MHz with burning bootloader after selecting e.g. ‘ATtiny45 (8 MHz)’ on Board menu.
(Or, it also can work with external 20MHz using a resonator or a crystal, I guess.)

See ‘Configuring the ATtiny to run at 8 MHz (for SoftwareSerial support)’ on this page.

Arduino – SoftwareSerial

NewSoftSerial | Arduiniana