John gets the first turn, so is asked the following question.

John knows the answer "TOPIARY".

If he had not known the word, that would have been the end of his turn.
John is awarded the letter tiles he needs to spell TOPIARY. He can either place the word TOPIARY itself on the board or instead another word using some or all of the letters in TOPIARY, so for example he could place PARTY or TRIP or PAIR or POT etc.

John decides to place the word TOPIARY itself on the board.
The first word must be placed horizontally as far to the left as possible whilst also covering the central square.

When John is happy with the placement of his letters, he finishes his turn which locks his letter tiles in place on the board and awards him 7 points for placing 7 letter tiles.

John's turn has now ended and it's Mary's turn. She is asked the following question:

Mary answers "Psychiatrist" correctly and so is awarded its letter tiles.
She now has to decided how to play PSYCHIATRIST. She either has to place the word PSYCHIATRIST on the board or, if she prefers, another word using some or all of the letters in PSYCHIATRIST.

Any word that she places on the board must interconnect like in a crossword with the word already on the board. After some thought, she decides to play PSYCHIATRIST itself like this:

Mary didn't have to use the I of TOPIARY to play PSYCHIATRIST; she could have used the T or the A or the R instead.
She could also have formed another word using letters from PSYCHIATRIST that interlock crossword-style with a letter in TOPIARY. There are lots of such words: THROAT, IS, SHIP, STAIRS, CHARITY, ARTISTIC, PHYSICIST, HAT, TORCH etc.
Once she had chosen how to place her letter tiles, Mary presses the button to finish her turn.

Note that the I of TOPIARY is now red. A letter that belongs to two words is the colour of the word more recently placed on the board. This is important because the aim of the game is to get 60 letters of your colour on the board: Mary is hoping that she will get 60 red letters on the board before John get 60 of his black letters on the board.
Mary's score has been updated to 12 points as she now has 12 letter tiles on the board; John's score has decreased to 6 as he lost the letter I in TOPIARY.
Now it's John's turn again:

John knows the correct word is SIDEWALK and so is awarded the word's letters.

He places all of the letter tiles on the board like this:

John now finishes his turn.

All of the letters of the word PSYCHIATRISTS are now black; the reason is that John has turned Mary's word PSYCHIATRIST (the letters of which were in red because Mary played it) into the new word PSYCHIATRISTS. All three of the words on the board are now in black because John made each of them. Notice that the first I in PSYCHIATRISTS has changed colour again: first (when John played TOPIARY) it was black, then when Mary played PSYCHIATRIST it was red, and now that John has made PSYCHIATRISTS it is black again.
It now looks at this point as if John is doing very well: he's got 26 of his black letters on the board and Mary now has none of her red letters on the board. But that can soon change...

On the next turn Mary's question is "Which branch of philosophy has been likened to a blind man in a dark room looking for a black cat that isn't there?" which she answers correctly with METAPHYSICS.

Again, Mary doesn't have to play METAPHYSICS itself; she could play THEM or MEATY or MATHS or PHYSICS or CAST etc. — indeed, any word made from some or all of the letters in METAPHYSICS that interlocks crossword-style with a word or words already on the board.

She decides to play METAPHYSICS itself and the board now looks like this:

At this point John has 18 of his black letters on the board and Mary has 19 of her red letters on the board.
It's now John's turn. His question is:

John knows the answer: BLUEBELLS.
It's not possible to place the word BLUEBELLS anywhere on the board so that it interlocks with another word. So John now has no choice but to try to make a word using some or all of the letters of BLUEBELLS.

After some thought John places BELLS on the board:

By playing BELLS like this, John now has 23 of his letters on the board, while Mary has lost one of her letters and now has 18:

Now it's Mary's turn. Her question is:

She knows the answer: EXULTATION.

After some thought, Mary decides that she's going to play the word EXULTATION rather than a word using letters from EXULTATION. There are two ways in which EXULTATION can fit on the board: using the first T of PSYCHIATRISTS or the T of TOPIARY. Mary can choose which she uses. She decides to use the T of TOPIARY:

The board now looks like this:

The game continues until the board looks like this:

Here John has just played NONE (in the column on the far right of the board) and now has 61 of his black letters on the board. If there are still at least 60 black letters on the board after Mary's turn John will have won the game.
But, Mary wins the word BROOM: